Kirby Problem 1.48

Definitions and mathematical statement

A knot $K\subset S^3$ is smoothly doubly slice if an unknotted smooth sphere $S^2\subset S^4$ intersects an equatorial $S^3$ transversely in $K$. Unknotted means ambiently isotopic to the standard sphere. The topological version uses locally flat embeddings and topological isotopy. A smoothly slice knot bounds a smooth disk in $B^4$. Write $\#$ for connected sum of knots or of oriented manifolds, as appropriate.

The questions concern cancellation and torsion phenomena:

1. If $K$ and $K\#J$ are smoothly doubly slice, must $J$ be smoothly doubly slice?
2. Is there a knot $K$ that is smoothly slice and topologically doubly slice, but not smoothly doubly slice, while $K\#K$ is smoothly doubly slice?
3. If closed connected $3$-manifolds $M$ and $M\#N$ admit smooth embeddings in $S^4$, must $N$ also admit a smooth embedding in $S^4$?

Ask (a) and (c) separately also in the locally flat topological category. Double sliceness requires the two slice disks on opposite sides of the equator to form a single unknotted sphere. Separate sliceness of each half, or ordinary concordance to a doubly slice knot, does not supply that condition.

Short English statement

Does double sliceness cancel under connected sum, can doubling remove its smooth obstruction, and does an analogous cancellation hold for $3$-manifolds in $S^4$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2707 (KP-1.48), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.48. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.